Profit Maximization and Returns to Scale

Assume a labor-intensive production function:

$$Q(L)=L^\beta$$

Find demand of labor that maximizes profits (unconditional demand $$L(p,w,r)$$), the demand of labor that minimizes costs (unconditional demand $$L(q,w,r))$$, the supply function $$(Q(L(p,w,r))$$), and the benefits $$(\pi(L(p,w,r))$$ if:

a) $$\beta>1$$

b) $$\beta=1$$

For the first case (increasing returns to scale) I am able to Maximize profits by getting the First Order Conditions, etc., and finding $$L(p,w,r))$$:

$$L(p,w,r))=(w/p\beta)^{1/\beta-1}$$

Thus, by replacing $$L(p,w,r))$$ into the benefits:

$$\pi(L(p,w,r)=p*(w/p\beta)^{\beta/\beta-1}-w*(w/p\beta)^{1/\beta-1}$$ (It can be simplified by doing more algebra).

By replacing $$L(p,w,r))$$ into the production function:

$$Q(L(p,w,r)=(w/p\beta)^{\beta/\beta-1}$$

Finally, the conditioned (on a production of "q") labor demand is given by:

$$L(q,w,r)=q^{1/\beta}$$

However, for the second case (constant returns to scale) I am not able to get the First Order Conditions and so on for obtaining the benefits, the supply function, etc. Thus, I would like to know how to address this problem.

I would appreciate any help,

• You'll probably have to consider corner solutions for the case where $\beta=1$. Also, why is labor demand a function of $r$ (presumably rent) when capital is not required in production? – Herr K. Feb 13 at 17:19
• You're right! My bad, I just wrote the standard form but sure, it only depends on $w$ and $p$, in the case of non-conditioned labor demand. For a single-input production function, corner solutions just refer to the case when $L(p,w)=0$? – F. Sánchez Feb 13 at 19:58
• Depending on the ratio of $p/w$, corner solution could be $0$ or the upper bound on labor supply. – Herr K. Feb 13 at 20:14